Many of the armies of large empires of world history were this, for instance Xerxes' army and navy that invaded Greece during the second Graeco-Persian war included contingents from pretty much every nation from Egypt to the Black Sea, from Asia Minor to the Indian marches.
The British Empire forces in World War One. The United Kingdom of Great Britain and Ireland, Australia, Canada, Newfoundland, New Zealand, India and East, West and South Africa.
Bill Slim's Fourteenth Army in World War Two, "a huge foreign legion of what Attlee called 'the scrapings of the barrel' from half the nations under the sun," as George Mac Donald Fraser put it. At the time, India was not a unified country, but a conglomerate of different princely states and tribes - Sikhs, Baluchis, Jats, Rajputs, Punjabis, Dogras, and many others, all of which had their own regiments. The Indian Army also included units from all four countries of the UK, Gurkhas and Afghans. Fourteenth Army drew primarily from the Indian Army, but also included Burmese, more troops from Nepal, Anglo-Indians, two West African divisions with troops from Nigeria, Gambia, the Gold Coast and Sierra Leone, and an East African division with troops from Kenya, Tanganyika, Nyasaland, Uganda, and Rhodesia. Not to mention significant external support from Nationalist China and the United States, which provided most of the air power and a Chindit-trained infantry regiment, Merrill's Marauders.
The British Eighth Army had British, Scottish, South African, Australian, Indian and New Zealand divisions, and smaller units of Free French, Czechs, Greeks and Poles.
The modern British armed forces count as well, come to think of it; recruitment is open to citizens of all Commonwealth countries as well as Ireland, and EU citizens or other immigrants can join if they've been resident in the UK for seven years.
The Austrian/Austro-Hungarian Empire was very much this trope. Although there was a predominant German (and Hungarian) veneer, the Habsburg domains were home to various peoples, cultures and beliefs, all more or less united by their loyalty to the clan. By World War I, its military forces spoke multiple languages that part of an officer's job was to learn most if not all of them. Though at the time, this also gave the impression that, compared to Imperial Germany, Austrians weren't exactly "pure" Germans.
Even more pronounced in the case of military alliances composed of several independent nations.
In the Seven Years' War the Prussian army and its allies (Great Britain and a few small German states) faced the armies of Austria, France, Russia, Sweden, Saxony, several smaller German states and of the Holy Roman Empire. That last one was in itself a microcosm of contingents supplied by the middling-to-minute principalities etc. of the Empire, each of which was a sovereign nation at the time.
The Allied and Associated Powers of World War One - France, Serbia, Russia, the British Empire, Belgium, Portugal, Japan, Italy, Romania, the United States etc. etc.
The 1991 Gulf War coalition, besides the US, Saudi Arabia and the UK, included over 20 other countries. Likewise, the 2003-2009 US/British-led Multinational Force in Iraq included around 40 countries during it's existence.
Although many of the armies from the Renaissance to at least the Thirty Years War were composed of soldiers of fortune from all over Europe in a similar way.
The planned Multilateral Force. Subs and Warships manned by multinational crews and equipped with nukes.
Many space missions, especially after the Cold War ended. Personnel from dozens of different countries have crewed Russian and American vessels side by side, not to mention the ISS, which is owned and operated by fifteen nations, and also the European Space Agency (18 nations). Major future missions, such as human travel to Mars, are likely to be international as they would be so expensive, and the need for Cold War one-upmanship has vanished.
Basically the point of Up With People. It's common to see upwards of fifteen different countries represented in a single cast. They also double as True Companions for the cast itself.
ESL programs often contain English teachers of various countries. More often than not, they are from countries in which English is an official language along with whatever host country has brought them. Japan, South Korea, Vietnam, and other countries have such programs.
Many modern charity organisations such as Amnesty International and the Red Cross count as this, being huge teams of like-minded idealists from across the globe.
Professional sports teams can end up this way, especially in high-profile leagues whose sports have wide international following (notable examples include the English Premier League, the National Basketball Association, and the National Hockey League). Such teams are often the most well-known and richest in their leagues, as they can find and attract both overseas talent and fans.